⚛️ high-energy theory

Electromagnetic Duality Sensitivity of Holographic Complexity

This paper demonstrates that within Einstein-ModMax theory, certain matter-sensitive holographic complexity functionals can distinguish between electromagnetic configurations that are indistinguishable by gravitational invariants alone, thereby revealing how different choices of complexity functionals retain varying degrees of sensitivity to bulk matter information.

Original authors: Mojtaba Shahbazi, Mehdi Sadeghi

Published 2026-09-15

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CC BY 4.0

Original authors: Mojtaba Shahbazi, Mehdi Sadeghi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the deepest reaches of theoretical physics, scientists use a concept called holography to understand how the universe works. Imagine a three-dimensional space, like the interior of a star or a black hole, that is completely described by information living on its two-dimensional surface, much like a hologram on a credit card contains the image of a 3D object. In this framework, the "complexity" of a system is not just a measure of how many parts it has, but a measure of how difficult it is to build that system from scratch. Physicists have long suspected that this complexity is tied to the shape of space itself. However, a newer idea suggests that complexity is not limited to just the shape of space; it can also depend on the invisible fields that fill that space, such as electricity and magnetism. This raises a fascinating question: if two different arrangements of electric and magnetic fields create the exact same shape of space, can a measurement of complexity tell them apart?

A team of researchers at Ayatollah Boroujerdi University in Iran set out to answer this question by studying a specific type of theoretical universe where gravity and electromagnetism are tightly linked. They focused on a special kind of matter that behaves in a unique way: it treats electric and magnetic fields as two sides of the same coin, allowing them to be rotated into one another without changing the underlying laws of physics. In their model, they started with a black hole that was charged only with electricity. Because of the special symmetry of the matter they were studying, they could mathematically rotate this electric charge into a mix of electric and magnetic charge, or even a purely magnetic charge, without altering the geometry of the black hole or the way it pulls on nearby objects. To an observer looking only at the shape of space or the force of gravity, these different configurations would look exactly the same.

The researchers then applied the "complexity equals anything" framework to these scenarios. This approach allows scientists to define complexity using different mathematical tools. Some tools look only at the curvature of space, while others look at the specific matter fields filling that space. When the team used a tool that looked only at the shape of space, the complexity remained exactly the same, regardless of whether the black hole was electric, magnetic, or a mix of both. This confirmed that the gravitational geometry alone cannot distinguish between these different electromagnetic states. However, when they used a tool that was sensitive to the electromagnetic fields themselves, the results changed dramatically. The calculated complexity and the rate at which it grew over time shifted as they rotated the electric field into a magnetic one.

This finding reveals that the choice of how to measure complexity is not just a technical detail; it determines what information the measurement can see. The study showed that while the gravitational structure of the universe might be blind to the specific mix of electric and magnetic charges, a complexity measurement that pays attention to matter can detect the difference. The researchers found that as they rotated the fields, the behavior of the complexity growth rate could change in surprising ways. For certain angles of rotation, the mathematical structure describing the growth rate would shift, creating new peaks and valleys in its behavior. They identified specific critical angles, near 61 degrees and 118 degrees, where this structural change occurred, marking a transition in how the complexity evolves.

From the perspective of the boundary of this universe, the story is equally revealing. Even though the shape of space and the energy-momentum tensor remained identical for all the different electromagnetic configurations, the data describing the electromagnetic fields at the edge of the universe changed. A rotation that started with a purely electric setup generated magnetic data at the boundary. This means that two states that look identical in terms of gravity and energy are actually different quantum states. The study demonstrates that generalized holographic complexity has the unique ability to act as a diagnostic tool, capable of distinguishing between these hidden differences. By choosing the right functional, or measurement tool, scientists can access information about the matter sector that gravity alone keeps hidden. This work provides a concrete example of how the freedom to choose different definitions of complexity allows physicists to probe different layers of reality, showing that what we measure depends fundamentally on how we choose to look.

Technical Summary: Electromagnetic Duality Sensitivity of Holographic Complexity

Problem Statement
The paper investigates whether holographic complexity can distinguish between electromagnetic configurations that are related by electromagnetic duality but share identical gravitational geometries and energy-momentum tensors. In Einstein-ModMax theory—a nonlinear extension of Maxwell theory that preserves both conformal invariance and electromagnetic duality—continuous duality rotations transform purely electric configurations into mixed electric-magnetic or purely magnetic ones without altering the bulk metric or the stress-energy tensor. While standard holographic complexity proposals (CV, CA, CV2.0) rely on geometric quantities that remain invariant under such transformations, the "Complexity=Anything" (CAny) framework allows for a broader class of functionals. The central question is whether the functional freedom inherent in CAny allows for the construction of observables sensitive to the underlying electromagnetic matter sector, even when the gravitational sector is blind to it.

Methodology
The authors utilize the Einstein-ModMax theory in a four-dimensional Anti-de Sitter (AdS) background. They construct a static, planar AdS black brane solution with a purely electric charge.

Duality Orbit Construction: They apply the electromagnetic duality transformation, an SO(2) rotation parameterized by an angle $\theta$ , to the electromagnetic field strength tensor. This generates a family of solutions (the duality orbit) ranging from purely electric ( $\theta=0$ ) to mixed and purely magnetic ( $\theta=\pi/2$ ) configurations. Crucially, the metric function $f(r)$ and the stress-energy tensor remain invariant along this orbit.
Complexity Functional Analysis: Within the CAny framework, they evaluate two distinct types of complexity functionals:
- R-invariant functionals: Constructed solely from gravitational curvature invariants (e.g., the Weyl tensor squared, $C^2$ ).
- R-variant (matter-sensitive) functionals: Constructed using electromagnetic invariants, specifically $F_{\mu\nu}F^{\mu\nu}$ .
Calculation of Complexity and Growth: The authors calculate the generalized holographic complexity and its growth rate ( $dC/d\tau$ ) for the black brane solution. They analyze the effective potential $\tilde{U}(r)$ governing the extremization of the codimension-one hypersurface.
Orbit Complexity Definition: They define "orbit complexity" ( $\Delta R C_{Any}$ ) as the difference in complexity between a rotated configuration and the original electric configuration. They also introduce a "duality sensitivity" parameter to quantify the response of the functional to the rotation angle $\theta$ .
Numerical and Analytical Investigation: The study involves numerical analysis of the complexity growth rate and an analytical derivation of critical duality angles ( $\theta_c$ ) where the structure of the effective potential changes, specifically where the number of extrema (and thus the number of branches in the complexity growth rate) changes.

Key Contributions and Results

Duality Sensitivity of Functionals: The paper demonstrates that the sensitivity of holographic complexity to electromagnetic duality is entirely dependent on the choice of the functional $F_1$ $F_{1}$ .
- Functionals based on curvature invariants (e.g., $1 + \alpha C^2$ ) are R-invariant. Their values and growth rates remain constant along the duality orbit because the metric is unchanged.
- Functionals based on matter invariants (e.g., $1 + \alpha F_{\mu\nu}F^{\mu\nu}$ ) are R-variant. Their values and growth rates vary non-trivially with the duality angle $\theta$ , interpolating between electric, mixed, and magnetic configurations.
Orbit Complexity and Critical Angles: The authors show that for matter-sensitive functionals, the effective potential $\tilde{U}(r)$ governing the complexity growth can undergo structural changes as $\theta$ varies. Specifically, they identify critical duality angles (e.g., $\theta_c \approx 61^\circ$ and $118^\circ$ for specific parameters) where the number of local maxima in the potential changes. This leads to a transition in the number of branches of the complexity growth rate, a phenomenon not observed in curvature-based functionals.
Boundary Interpretation: From the boundary perspective, the duality rotation preserves the boundary metric and the holographic stress tensor. However, it rotates the asymptotic electromagnetic data (generating magnetic fields from electric ones). The paper argues that matter-sensitive complexity functionals can distinguish these boundary states, whereas curvature-based functionals cannot. This implies that the "freedom" in choosing a complexity functional corresponds to a choice of which sector of bulk information (gravitational vs. matter) the observable probes.

Significance and Claims
The paper claims that electromagnetic duality serves as a diagnostic tool to classify generalized holographic complexity functionals based on their information content.

Physical Interpretation of Functional Freedom: The results provide a concrete physical realization of the "Complexity=Anything" proposal's functional freedom. The choice of functional is not merely a mathematical ambiguity but determines the specific sector of the bulk theory (gravitational vs. matter) that the complexity observable is sensitive to.
Distinguishing Degenerate Geometries: The study establishes that generalized holographic complexity can resolve ambiguities that standard geometric observables cannot. Two bulk configurations that are indistinguishable by their metric and stress tensor (degenerate in the gravitational sector) can be assigned different complexity values if a matter-sensitive functional is chosen.
Limitations and Scope: The authors modestly frame these results as an explicit realization within the Einstein-ModMax framework. They do not claim that all matter-sensitive functionals will exhibit critical branching behavior, but rather that the potential for such sensitivity exists and depends on the specific functional and parameters. The paper suggests that this duality sensitivity offers a criterion for organizing complexity functionals but does not propose new experimental applications or extend the results to non-Abelian systems or higher dimensions within this work.

Technical Summary: Electromagnetic Duality Sensitivity of Holographic Complexity

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